Problem 30 - Precalculus Mathematics for Calculus

Question

Find all rational zeros of the polynomial, and write the polynomial in factored form. $$P(x)=2 x^{4}-x^{3}-19 x^{2}+9 x+9$$

Step-by-Step Solution

Answer

The rational zeros are \( x = 1 \) and \( x = 3 \). Polynomial factored: \( (x - 1)(x - 3)(2x + 1)(x + 3) \).

Step 1: Rational Root Theorem

Use the Rational Root Theorem to identify potential rational zeros. The possible rational roots are the factors of the constant term (9) divided by the factors of the leading coefficient (2). So, the possible rational roots are \( \pm 1, \pm 3, \pm 9, \pm \frac{1}{2}, \pm \frac{3}{2}, \pm \frac{9}{2} \).

Step 2: Test Potential Rational Zeros

Substitute the potential rational zeros into the polynomial \( P(x) \) to find the actual zeros. After testing, you find that \( x = 1 \) is a root because \( P(1) = 0 \).

Step 3: Synthetic Division

Perform synthetic division on \( P(x) \) using \( x = 1 \) as a root. Divide \( P(x) \) by \( x-1 \) to get the quotient polynomial. The division results in \( 2x^3 + x^2 - 18x - 9 \).

Step 4: Factor the Quotient Polynomial

Factor \( 2x^3 + x^2 - 18x - 9 \) using methods like synthetic division or polynomial factoring. Testing another rational root, you find that \( x = 3 \) is also a root. Dividing yields the quotient \( 2x^2 + 7x + 3 \).

Step 5: Factor the Quadratic

Factor the quadratic \( 2x^2 + 7x + 3 \). This can be factored into \((2x + 1)(x + 3)\) using techniques such as trial and error or the quadratic formula.

Step 6: Complete the Factored Form

Combine all factors to express the original polynomial in its completely factored form: \( P(x) = (x - 1)(x - 3)(2x + 1)(x + 3) \).

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a powerful tool for finding potential rational zeros of a polynomial. Rational zeros are fractions or whole numbers that make the polynomial equal to zero. To apply the theorem, you need two things:

• The factors of the constant term (last number) in the polynomial.
• The factors of the leading coefficient (number in front of the highest power of x).

For example, given the polynomial \( P(x)=2x^{4}-x^{3}-19x^{2}+9x+9 \):

• The constant term is 9, so its factors are \( \pm 1, \pm 3, \pm 9 \).
• The leading coefficient is 2, so its factors are \( \pm 1, \pm 2 \).

The possible rational zeros are all the combinations of these formed by dividing factors of the constant term by factors of the leading coefficient: \( \pm 1, \pm 3, \pm 9, \pm \frac{1}{2}, \pm \frac{3}{2}, \pm \frac{9}{2} \).
The next step is to test these possibilities to find which ones actually make the polynomial zero.

Synthetic Division

Synthetic division is a quick and efficient method for dividing a polynomial by a linear factor, generally of the form \( x - c \) where \( c \) is a potential rational root. You use synthetic division to check if a number (let's say \( x = 1 \)) is actually a root of the polynomial.
Here's how you perform synthetic division with \( x = 1 \) as a potential root for our polynomial \( P(x) \):

• Write down the coefficients of the polynomial: \(2, -1, -19, 9, 9\).
• Place the potential root \( c = 1 \) to the left.
• Bring the first coefficient down directly.
• Multiply \( c \) by this number and add it to the next coefficient. Repeat across all coefficients.

The remainder of this process should be zero if \( c \) is an actual root. This method yields a simplified polynomial expression. If it’s not zero, \( c \) is not a root. With \( x = 1 \), the remainder is zero, verifying it as a root. This allows you to simplify the polynomial to \( 2x^3 + x^2 - 18x - 9 \). This new polynomial is then used to check for further roots.

Polynomial Factoring

After narrowing down possible rational roots using the Rational Root Theorem and synthetic division, you often need to factor the polynomial further. This process involves expressing the polynomial as a product of simpler polynomial factors.
Once we found \( x=1 \) as a root and divided the polynomial, we were left with \( 2x^3 + x^2 - 18x - 9 \). By testing more potential rational roots, we discovered \( x = 3 \) is another root. Performing synthetic division again simplifies it to \( 2x^2 + 7x + 3 \).
The resulting quadratic polynomial can be factored using different methods such as:

• Trial and error: Trying combinations that might multiply to give the constant term while adding up to the coefficient of \( x \).
• Quadratic formula: For general solutions to any quadratic equation.

Here, \( 2x^2 + 7x + 3 \) factors neatly to \((2x + 1)(x + 3)\).Finally, incorporate all the factors found, creating the fully factored form for \( P(x) = (x - 1)(x - 3)(2x + 1)(x + 3) \). This expression provides a complete breakdown of the polynomial into its simplest forms, giving all the zeros clearly. This step successfully wraps up the factoring process.